Eilenberg--Mac Lane space

Let $n\ge 1$.

• For an arbitrary group $G$, an Eilenberg--Mac Lane space of type $K(G,1)$ is a connected based CW complex $K$ equipped with a specified isomorphism ${\pi }_1(K)\cong G$ and satisfying ${\pi }_i(K)=0$ for every $i>1$.
• For an abelian group $A$ and $n\ge 2$, an Eilenberg--Mac Lane space of type $K(A,n)$ is a connected based CW complex $K$ equipped with a specified isomorphism ${\pi }_n(K)\cong A$ and satisfying ${\pi }_i(K)=0$ for every positive $i\mathrm{\ne }n$.

The notation $K(G,n)$ denotes a chosen model together with its chosen group identification, not a literally unique space. No $K(G,n)$ with nonabelian $G$ and $n\ge 2$ is asserted: higher homotopy groups are abelian. The zero-group case is allowed and has the homotopy type of a point.